Is Zakat now due on the Zakat money that was with my father, may Allah have mercy on him, and upon which a year has passed?

Delaying obligatory Zakat (definition) without a valid excuse is a mistake and a sin. Whoever is obligated to pay Zakat is not permitted to delay it; rather, it must be distributed immediately. It is not permissible for the owner to keep it to distribute throughout the year. Therefore, the remaining Zakat amount must be disbursed immediately, and it is not subject to Zakat again, even if a year passes before its distribution.